Yes I think I see what you are saying . I think of the two functions as moving in opposite directions. I did some pondering on this matter today while trying to come to grips with this concept of inverse functions. I finally concluded this: If a function is like a black box into which you put, let's say, a piece of raw chocolate and out of which pops a bon bon. Then the inverse of that is a black box into which you put the bon bon the other function made and out pops a piece of raw chocolate. Really, the concept is not hard to grasp but the notation used to indicate what is or has happened is so, I don't know...so arcane? or maybe non-intuitive...that I keep losing the thread of what came from where and where it has to go to get back where it came from. I confess that this business of explaining these ideas using nothing but a and b and f(x) and g(x) and h(x) and all the rest without, at least for the duration of the explanation, presenting any actual examples....well, it's not the way I would do it. I guess these textbook authors get so used to this language that they forget how alien it feels to those who are relatively new to it.
The notion itself is very simple, as simple as a list containing two columns. In the left column are listed all the members of some class, say the names of the students in a class, with each member listed once. In the right column are listed members of some class, say seat numbers in an auditorium. Each row gives the item in the right column that is uniquely associated with the item in the left column. In our example, the assignment of specific students to specfic seats is a function. In other words, the basic notion is one of association.
The notion of a function is a bit more complex than mere association because it demands unique and exhaustive association in one direction. Another example will make this clearer. Suppose we have three rooms, each of which contains only people who are devout members of some religion and are joined to some other person in the room in a marriage recognized as valid by that religion.
In one room, the religion universally subscribed to accepts only polygyny as a valid marriage. In that room, you can find a function associating
each woman to her
unique husband, but you cannot create a function associating each man to his unique wife because each man has more than one wife.
In another room, the religion universally subscribed to accepts only polyandry as a valid marriage. In that room, you can find a function associating
each man to his
unique wife, but you cannot find a function associating each woman to her unique husband because each woman has more than one husband.
In the third room, the religion universally subscribed to accepts only monogamy between a biological man and a biolgical woman as a valid marriage. In that room, you can find a function associating
each man with his
unique wife. Moreover, you can find a different function associating
each woman with her
unique husband. These are inverse functions because if you apply them one after the other, you end up with the person you started with.
The whole idea of a function is to make
exact a very important kind of association, one that is exhaustive and unique in one direction. Getting an idea formulated exactly frequently leads to a very abstract definition that seems far indeed from the familiar idea at its root. Then we create a very simple notation in which to pack these abstractions.
